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262 Polynomial Approximation of Differential Equations
an operator, that for any pair of input values (yk−1,yk) in (11.3.1) provides the out-
put values (q ′ n,k (sk−1),q ′ n,k (sk)). For f ≡ 0, σ1 = σ2 = 0, the mapping Γ is
linear and corresponds to the Schur complement of the matrix associated with (11.2.4)-
(11.2.7), with respect to the unknowns corresponding to the interface points (see duff,
erisman and reid(1986), p.60). Several iterative techniques are available for the nu-
merical determination of the zeroes of Γ. Convergence results in an abstract context
are given in agoshkov(1988). The conjugate gradient method, which is theoretically
examined in the case of finite element multidomain approximations in bjørstad and
widlund(1986), is studied in chan and goovaerts(1989) for spectral Legendre ap-
proximations. We note that the rate of convergence of the method depends on the ratio
between the maximum and the minimum size of the domains.
Variants of the algorithm proposed above have been analyzed recently. An itera-
tive approach is obtained by alternating Dirichlet and Neumann boundary conditions
in the solution of the problems in the different subdomains. The solution in a given
domain, recovered by imposing Dirichlet (respectively Neumann) boundary conditions,
provides the Neumann (respectively Dirichlet) data to be used in the contiguous do-
mains. Although only half of the blocks can be simultaneously processed at any step, a
speed up of the convergence is observed when using an appropriate relaxation scheme.
The procedure has been investigated in zanolli(1987), funaro, quarteroni and
zanolli(1988), and interpreted in quarteroni and sacchi-landriani(1988) as a pre-
conditioned iterative method for the Schur complement of the matrix corresponding to
(11.2.4)-(11.2.7). Other results are provided in zampieri(1989). A similar algorithm
for the discretization of first-order time-dependent differential problems is analyzed in
quarteroni(1990).
The interest in applying domain-decomposition methods, for the solution of partial
differential equations, has increased in recent years, together with the development of
parallel computers. Therefore, many other sophisticated techniques have been experi-
mented. The reader will find a collection of papers and references in chan, glowinski,
periaux and widlund(1989) and chan, glowinski, periaux and widlund(1990).